4.3 Transformations & similarity

Transformations, Congruence, and Similarity

A transformation moves or resizes a figure on the coordinate plane. Three transformations — translations, reflections, and rotations — are rigid motions: they slide, flip, or turn a figure without changing its size or shape, so the image is congruent to the original. A fourth, dilation, shrinks or enlarges a figure, producing an image that is similar but usually not congruent.

The four transformations

Translations

A translation adds the same amount to every point's coordinates. The rule (x, y) → (x + 4, y − 3) slides a figure 4 units right and 3 units down. Worked example: the point (2, 5) becomes (2 + 4, 5 − 3) = (6, 2). Because every vertex moves identically, side lengths and angles are preserved.

Reflections

A reflection flips a figure across a line. Two common rules are: across the x-axis, (x, y) → (x, −y); and across the y-axis, (x, y) → (−x, y). Worked example: reflecting (3, 4) across the x-axis gives (3, −4). The image sits the same distance from the mirror line as the original, on the opposite side.

Rotations

A rotation turns a figure about a point, usually the origin. A 90° counterclockwise turn uses (x, y) → (−y, x); a 180° turn uses (x, y) → (−x, −y). Worked example: rotating (4, 1) by 180° about the origin gives (−4, −1). Angles and lengths stay the same throughout.

Dilations and scale factor

A dilation multiplies each coordinate by a scale factor k, measured from the origin: (x, y) → (kx, ky). If k > 1 the figure grows; if 0 < k < 1 it shrinks. Worked example: dilating (6, 8) by k = ½ gives (½ · 6, ½ · 8) = (3, 4). The image is similar to the original: every length is multiplied by k, while every angle is unchanged.

Congruence versus similarity

Congruent figures have the same shape and the same size — corresponding sides are equal and corresponding angles are equal (symbol ≅). Similar figures have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are proportional (symbol ~). Every congruent pair is also similar (with scale factor 1), but not every similar pair is congruent.

Similar-triangle proportions

When two triangles are similar, matching sides form equal ratios. If △ABC ~ △DEF, then AB/DE = BC/EF = AC/DF. Setting two of these ratios equal creates a proportion you solve by cross-multiplication.

Worked example: △ABC ~ △DEF with AB = 6, BC = 9, and the corresponding side DE = 4. Find EF.

Set up AB/DE = BC/EF → 6/4 = 9/EF.

Cross-multiply: 6 · EF = 4 · 9 = 36, so EF = 36 / 6 = 6.

Worked example (scale factor): the ratio of corresponding sides is DE/AB = 4/6 = 2/3, so triangle DEF is a two-thirds-size copy of ABC. Multiplying any side of ABC by 2/3 gives the matching side of DEF: 9 × 2/3 = 6, matching the EF found above.

Indirect measurement

Similar triangles let you measure things you cannot reach, such as a flagpole's height using shadows. If a 6 ft person casts a 4 ft shadow while a flagpole casts a 20 ft shadow at the same moment, the two triangles are similar because the sun's rays make equal angles:

height / shadow is constant → 6/4 = h/20.

Cross-multiply: 4h = 6 × 20 = 120, so h = 120 / 4 = 30 ft.

Identifying a transformation from coordinates

Sometimes you are shown an original figure and its image and asked to name the transformation. Compare the coordinates point by point: if every point shifted by the same amounts, it is a translation; if exactly one coordinate flipped its sign, it is a reflection; if both coordinates flipped sign, it is a 180° rotation; and if every coordinate was multiplied by the same number, it is a dilation.

Worked example: A(1, 2) and B(3, 4) map to A'(2, 4) and B'(6, 8). Each coordinate doubled, so this is a dilation with scale factor k = 2. The image is similar to the original — twice as large with identical angles. Because k > 1, the figure grew, which matches the larger image coordinates and passes a quick reasonableness check.

Reasonableness and key ideas

A scale factor greater than 1 must give a bigger image; a factor between 0 and 1 must give a smaller one — if your dilated figure changed size in the wrong direction, recheck k. Rigid motions never change measurements, so a "translated" figure with different side lengths signals an arithmetic slip. And in any proportion, matching parts must sit in matching positions — for instance, all originals on top and all images on the bottom. Mixing them up is the single most common similar-triangle error on the test.